hidden symmetry and quantum phases in spin 3/2 cold atomic systems
DESCRIPTION
Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems. Congjun Wu. Kavli Institute for Theoretical Physics, UCSB. Ref: C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003); C. Wu, Phys. Rev. Lett. 95, 266404 (2005); - PowerPoint PPT PresentationTRANSCRIPT
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Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
Ref: C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003);
C. Wu, Phys. Rev. Lett. 95, 266404 (2005);
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005);
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
OSU , 05/09/2006
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Collaborators
• S. Chen, Institute of Physics, Chinese Academy of Sciences, Beijing.• J. P. Hu, Purdue.
• S. C. Zhang, Stanford.
• Y. P. Wang, Institute of Physics, Chinese Academy of Sciences, Beijing.
• L. Balents, UCSB.
Many thanks to E. Demler, M. P. A. Fisher, E. Fradkin, T. L. Ho, A. J. Leggett, D. Scalapino, J. Zaanen, and F. Zhou for very helpful discussions.
• O. Motrunich, UCSB
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Rapid progress in cold atomic physics
M. Greiner et al., Nature 415, 39 (2002).
• Magnetic traps: spin degrees of freedom are frozen.
• In optical lattices, interaction effects are adjustable. New opportunity to study strongly correlated high spin systems.
M. Greiner et. al., PRL, 2001.
• Optical traps and lattices: spin degrees of freedom are released; a controllable way to study high spin physics.
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• Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
High spin physics with cold atoms
D. M. Stamper-Kurn et al., PRL 80, 2027 (1998); T. L. Ho, PRL 81, 742 (1998); F. Zhou, PRL 87, 80401 (2001); E. Demler and F. Zhou, PRL 88, 163001 (2002);T. L. Ho and S. Yip, PRL 82, 247 (1999); S. Yip and T. L. Ho, PRA 59, 4653(1999).
• Most atoms have high hyperfine spin multiplets.
• High spin fermions: zero sounds and Cooper pairing structures.
F= I (nuclear spin) + S (electron spin).
Spin-3/2: 132Cs, 9Be, 135Ba, 137Ba, 201Hg.
Spin-5/2 : 173Yb (quantum degeneracy has been achieved)
• Next focus: high spin fermions.
Fukuhara et al., cond-mat/06072
28
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Difference from high-spin transition metal compounds
• New feature: counter-intuitively, large spin here means large quantum fluctuations.
eee
high-spin transition metal compounds
high-spin cold atoms
eee
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Hidden symmetry: spin-3/2 atomic systems are special!
• Hidden SO(5) symmetry without fine tuning! Continuum model (s-wave scattering); the lattice-Hubbard model.
Exact symmetry regardless of the dimensionality, lattice geometry, impurity potentials.
SO(5) in spin 3/2 systems SU(2) in spin ½ systems
• This SO(5) symmetry is qualitatively different from the SO(5) theory of high Tc superconductivity.
• Spin 3/2 atoms: 132Cs, 9Be, 135Ba, 137Ba, 201Hg.
Brief review: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006)
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What is SO(5)/Sp(4) group?
,
• SO(3) /SU(2) group.
.,, 312312 LLLLLL yxz
3-vector: x, y, z.
3-generator:
2-spinor:
• SO(5) /Sp(4) group.
)51( baLab
5-vector:
10-generator:
4-spinor:
54321 ,,,, nnnnn
2
3
2
1
2
1
2
3
21, nn43 , nn
5n
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Quintet superfluidity and half-quantum vortex
• High symmetries do not occur frequently in nature, since every new symmetry brings with itself a possibility of new physics, they deserve attention.
---D. Controzzi and A. M. Tsvelik, cond-mat/0510505• Cooper pairing with Spair=2.
• Half-quantum vortex (non-Abelian Alice string) and quantum entanglement.
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
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Multi-particle clustering order
• Feshbach resonances: Cooper pairing superfluidity.
• Driven by logic, it is natural to expect the quartetting order as a possible focus for the future research.
• Quartetting order in spin 3/2 systems.
k 1
k 1
k 2
k 2
C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
4-fermion counter-part of Cooper pairing.
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Strong quantum fluctuations in magnetic systems
• Intuitively, quantum fluctuations are weak in high spin systems.
From Auerbach’s book:
S
S
Si
S
S
S
S kijk
ji 1],[
• However, due to the high SO(5) symmetry, quantum fluctuations here are even stronger than those in spin ½ systems.
large N (N=4) v.s. large S.
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Outline
• The proof of the exact SO(5) symmetry.
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, 186402(2003).
• Quintet superfluids and non-Abelian topological defects.
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• Not accidental! 1/r Coulomb potential gives rise to a hidden conserved quantity.
• An obvious SO(3) symmetry: (angular momentum) .
The SO(4) symmetry in Hydrogen atoms
• Q: What are the hidden conserved quantities in spin 3/2 systems?
L
• The energy level degeneracy n2 is mysterious.
H
A
L
A
L
A
A ,L .Runge-Lentz vector ; SO(4) generators
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Generic spin-3/2 Hamiltonian in the continuum
5~1
20
22
2/1,2/3
)()(2
)()(2
)()ˆ2
()(
aaa
d
rrg
rrg
rm
rrdH
• Pauli’s exclusion principle: only Ftot=0, 2
are allowed; Ftot=1, 3 are forbidden.
• The s-wave scattering interactions and spin SU(2) symmetry.
2
3
2
1
2
1
2
3
(r )
00 | 3
2
3
2;
(r )
(r )
a(r )
2a | 3
2
3
2;
(r )
(r )quintet:
singlet:
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Spin-3/2 Hubbard model in optical lattices
• The single band Hubbard model is valid away from resonances.
5~120
,,,,
,
)()()()(
.}.{
aaa
i
ii
ijij
i
iiUiiU
ccchcctH
U0.2
E 0.1,
l0 ~ 5000A, as,0,2 ~ 100aB
(V0
E r
)1/ 4 1 ~ 2as: scattering length, Er: recoil energy
E
t
U0,20V
0l
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SO(5) symmetry: the single site problem
00 E
45 205 UUE
322
502
14 UUE
203 UE
222 UE
;
1E
,
,
,
;
,
,
;
;
,
,
;
,
,
,
,
,
,
,
,
.
SU(2) SO(5) degen
eracy
E0,3,5 singlet
scalar 1
E1,4 quartet spinor 4
E2 quintet vector 5
)1(2int nnU
H
• U0=U2=U, SU(4) symmetry.
24=16 states.
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1,
Fx,Fy,Fz
ijaFiF j (a1 ~ 5) :
ijka FiF jFk (a1 ~ 7)
Particle-hole bi-linears (I)
• Total degrees of freedom: 42=16=1+3+5+7.
1 density operator and 3 spin operators are not complete.
rank: 0 1
2
3
• Spin-nematic matrices (rank-2 tensors) form five- matrices (SO(5) vector) .
)5,1(,2},{, baFF abba
jiaij
a
M
Fx2 Fy
2, Fz2 5
4,
{Fx,Fy},{Fy,Fz}, {Fz,Fx}M
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Particle-hole channel algebra (II)
• Both commute with Hamiltonian. 10 generators of SO(5): 10=3+7.
7 spin cubic tensors are the hidden conserved quantities.
ab i
2[a ,b ] (1a b5)
• SO(5): 1 scalar + 5 vectors + 10 generators = 16
kjiaijkzyx FFFF and,,
;
;
;
2
1
2
1
ab
a
ab
a
L
n
n
1 density:
5 spin nematic:
3 spins + 7 spin cubic tensors:
Time Reversal
even
even
odd
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Outline
• The proof of the exact SO(5) symmetry.
• Quintet superfluids and Half-quantum vortices.
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
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g2<0: s-wave quintet (Spair=2) pairing
• BCS theory: polar condensation; order parameter forms an SO(5) vector.
d 4S
ai
a de ˆ
Ho and Yip, PRL 82, 247 (1999);
Wu, Hu and Zhang, cond-mat/0512602.
5d unit vector
)(: 1xy rd
)(: 522 rdyx
i
)(: 2 rd xz i
)(: 3 rd yz
irdrz
)( : 43 22
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Superfluid with spin: half-quantum vortex (HQV)
,ˆˆ dd• Z2 gauge symmetry
• -disclination of as a HQV.
d
• Fundamental group of the manifold.
24
1 )( ZRP
244 /:ˆ ZSRPd
• is not a rigorous vector, but a directionless director.
ai
a de ˆ remains single-valued.
d
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Stability of half-quantum vortices
• Single quantum vortex:
L
M
hE sf log
4 2
})ˆ()({4
222
dM
drE spsf
}
{
log2
log24 2
R
L
L
M
hE
spsf
spsf
sfsp • Stability condition:
• A pair of HQV:
R
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Example: HQV as Alice string (3He-A phase)
• Example: 3He-A, triplet Cooper pairing.
22 /:ˆ ZSd
x
z
y
n
• A particle flips the sign of its spin after it encircles HQV.
A. S. Schwarz et al., Nucl. Phys. B 208, 141(1982);
F. A. Bais et al., Nucl. Phys. B 666, 243 (2003);M. G. Alford, et al. Nucl. Phys. B 384, 251
(1992).P. McGraw, Phys. Rev. D 50, 952 (1994).
Configuration space U(1)
d
ii ee ,
0,
,0)ˆ(
i
i
e
enU
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The HQV pair in 2D or HQV loop in 3D
P. McGraw, Phys. Rev. D, 50, 952 (1994).
x
z
y
n
zSO(3)SO(2)
• Phase state.
vortzvortiS 0)exp(
ie
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SO(2) Cheshire charge (3He-A)
• For each phase state, SO(2) symmetry is only broken in a finite region, so it should be dynamically restored.
vortvortimdm )exp(
Sz mvort
m mvort
• Cheshire charge state (Sz) v.s phase state.
vortvortdm 0
• HQV pair or loop can carry spin quantum number.
z
P. McGraw, Phys. Rev. D, 50, 952 (1994).
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Spin conservation by exciting Cheshire charge
• Initial state: particle spin up and HQV pair (loop) zero charge.
• Final state: particle spin down and HQV pair (loop) charge 1.
vortpvortpdminit 0
vortp
vort
i
p
m
edfinal
1
• Both the initial and final states are product states. No entanglement!
P. McGraw, Phys. Rev. D, 50, 952 (1994).
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Quintet pairing as a non-Abelian generalization
• HQV configuration space , SU(2) instead of U(1).
4e
n
S4
5,3,2,1e
4ˆ//)0(ˆ ed
nend ˆsinˆcos)ˆ,(ˆ242
vortn
equator: )2(ˆ 3 SUSn
-disclination
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SU(2) Berry phase
• After a particle moves around HQV, or passes a HQV pair:
2
1,2
1,
2
3,2
3
2351
5123)ˆ(
0
0
inninn
inninnnW
W
W 2
1
2
1
2
3
2
3 ,,
)2(3 SUS 5533
2211
ˆˆ
ˆˆˆ
enen
enenn
n
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Non-Abelian Cheshire charge
• Construct SO(4) Cheshire charge state for a HQV pair (loop) through S3 harmonic functions.
vortTTTTSnvortnnYndTTTT ˆ)ˆ(ˆ'';
333 '';
33
.,,)'( 1523,25,1335,12 LLLLLLTT
4e
• Zero charge state.
vortSnvortnnd ˆˆ00;00
3
SO(5) SO(4)
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Entanglement through non-Abelian Cheshire charge!
• SO(4) spin conservation. For simplicity, only
is shown.
init 3
2 p zero charge
vort
final 1
2 p Sz 1
vort 1
2 p Sz 2
vort
Sz T3 3
2T3
'
• Generation of entanglement between the particle and HQV loop!
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Outline
• The proof of the exact SO(5) symmetry.
• Alice string in quintet superfluids and non-Abelian Cheshire charge.
• SO(5) (Sp(4)) Magnetism in Mott-insulating phases.
• Quartetting and pairing orders in 1D systems.
C. Wu, Phys. Rev. Lett. 95, 266404(2005).
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Multiple-particle clustering (MPC) instability
• Pauli’s exclusion principle allows MPC. More than two particles form bound states.
SU(4) singlet:
4-body maximally entangled states
• Spin 3/2 systems: quartetting order.
)()()()( 2/32/12/12/3 rrrrOqt
• Difficulty: lack of a BCS type well-controlled mean field theory.
trial wavefunction in 3D: A. S. Stepanenko and J. M. F Gunn, cond-mat/9901317.
baryon (3-quark); alpha particle (2p+2n); bi-exciton (2e+2h)
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1D systems: strongly correlated but understandable
P. Schlottmann, J. Phys. Cond. Matt 6, 1359(1994).
• Competing orders in 1D spin 3/2 systems with SO(5) symmetry.
• Bethe ansatz results for 1D SU(2N) model:
2N particles form an SU(2N) singlet; Cooper pairing is not possible because 2 particles can not form an SU(2N) singlet.
Both quartetting and singlet Cooper pairing are allowed.
Transition between quartetting and Cooper pairing.
C. Wu, Phys. Rev. Lett. 95, 266404(2005).
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Phase diagram at incommensurate fillings
• Gapless charge sector.
• Spin gap phases B and C: pairing v.s. quartetting.
A: Luttinger liquid
20 gg
SU(4)
B: Quartetting
C: Singlet pairing
0g
2g
20 gg
SU(4)
• Singlet pairing in purely repulsive regime.
• Ising transition between B and C.
• Quintet pairing is not allowed.
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Competition between quartetting and pairing phases
• Phase locking problem in the two-band model.
.cos2
;cos24
21
21
ri
quar
ri
c
c
eO
e
1 3 / 2
3 / 2
2 1/ 2
1/ 2
overall phase; relative phase.
dual field of the relative phase
c r
A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).
• No symmetry breaking in the overall phase (charge) channel in 1D.
2..2
1,
2
3
2
1,
2
3 cc
i eie
r
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• Ising symmetry: .
Ising transition in the relative phase channel
2
1
2
1
2
3
2
3 , ii
quartquart OO ,
relative phase: rr rr
• Ising ordered phase:
Ising disordered phase:
)2cos2cos(2
1})(){(
2
121
22rrrxrxeff a
H
• the relative phase is pinned: pairing order;
the dual field is pinned: quartetting order.
21
21
Ising transition: two Majorana fermions with masses:
21
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Experiment setup and detection
M. Greiner et. al., PRL, 2001.
• Array of 1D optical tubes.
• RF spectroscopy to measure the excitation gap.pair breaking:
quartet breaking:
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Outline
• The proof of the exact SO(5) symmetry.
• Alice string in quintet superfluids and non-Abelian Cheshire charge.
• SO(5) (Sp(4)) magnetism in Mott-insulating phases.
• Quartetting v.s pairing orders in 1D spin 3/2 systems.
C. Wu, L. Balents, in preparation.
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005).
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Spin exchange (one particle per site)
• Spin exchange: bond singlet (J0), quintet (J2). No exchange in the triplet and septet channels.
• SO(5) or Sp(4) explicitly invariant form:
ijbaababex jnin
JJjLiL
JJH )()(
4
3)()(
42020
0,/4,/4
)()(
3122
202
0
2200
JJUtJUtJ
ijQJijQJHij
ex
3
23
2 0+2+1+
3
Lab: 3 spins + 7 spin cubic tensors; na: spin nematic operators; Lab and na together form the15 SU(4) generators.
5~1, ba
• Heisenberg model with bi-linear, bi-quadratic, bi-cubic terms.
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In a bipartite lattice, a particle-hole transformation on odd sites:
Two different SU(4) symmetries• A: J0=J2=J, SU(4) point.
Hex J {Lab (i)Lab ( j) na (i)nb ( j)}ij
• B: J2=0, the staggered SU’(4) point.
)(')()(')( jnjnjLjL abababab
ij
aaababex jninjLiLJ
H )}(')()(')({4
0
singlet quintet
triplet septet
J
singlet
quintet triplet septet
J0
*
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Construction of singlets
• The SU(2) singlet: 2 sites.
• The uniform SU(4) singlet: 4 sites.
• The staggered SU’(4) singlet: 2 sites.
baryon
meson
0)4()3()2()1(!4
)2()1(2
1 R
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Phase diagram in 1D lattice (one particle per site)
)4(20 SUJJ
spin gap dimer
gapless spin liquid
0J
2J
)4( 'SU
C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
• On the SU’(4) line, dimerized spin gap phase.• On the SU(4) line, gapless spin liquid phase.
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SU’(4) and SU(4) point at 2D
• J2>0, no conclusive results!
• At SU’(4) point (J2=0), QMC and large N give the Neel order, but the moment is tiny.
Read, and Sachdev, Nucl. Phys. B 316(1989).
K. Harada et. al. PRL 90, 117203, (2003).
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).
SU(4) point (J0=J2), 2D Plaquette order at the SU(4) point?
Exact diagonalization on a 4*4 lattice
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Exact result: SU(4) Majumdar-Ghosh ladder
• Exact dimer ground state in spin 1/2 M-G model.
H H i,i1,i2
i
, H i,i1,i2 J
2(S i
S i1
S i2)2
iJ
JJ2
1'
1i 2i
• SU(4) M-G: plaquette state.
cluster site-sixevery
iHH
SU(4) Casimir of the six-site cluster
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang,
Phys. Rev. B 72, 214428 (2005).
• Excitations as fractionalized domain walls.
sitessix
2
sitessix
2 )()( aabi nLH
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Conclusion
• Spin 3/2 cold atomic systems open up a new opportunity to study high symmetry and novel phases.
• Quintet Cooper pairing: the Alice string and topological generation of quantum entanglement.
• Quartetting order and its competition with the pairing order.
• Strong quantum fluctuations in spin 3/2 magnetic systems.
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SU(4) plaquette state: a four-site problem
• Bond spin singlet:
• Level crossing:
2
1
3
4
4!
0
• Plaquette SU(4) singlet:
J2
a b
J0
d-wave
s-wave
4-body EPR state; no bond orders
d-wave to s-wave
• Hint to 2D?
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Speculations: 2D phase diagram ?
• J2>0, no conclusive results!
• J2=0, Neel order at the SU’(4) point (QMC).
K. Harada et. al. PRL 90, 117203, (2003).
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).
2D Plaquette order at the SU(4) point?
Exact diagonalization on a 4*4 lattice
• Phase transitions as J0/J2? Dimer phases? Singlet or magnetic dimers?
J2
SU(4)
?
?
J0SU’(4)
C. Wu, L. Balents, in preparation.
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Strong quantum fluctuations: high symmetry
• Different from the transition metal high spin compounds!
SU(4) singlet plaquette state: 4 sites order without any bond order.
S. Chen, C. Wu, Y. P. Wang, S. C. Zhang, cond-mat/0507234, accepted by PRB.
classical Neel order
SU(2): S=3/2
…
large N large S quantum: spin disordered states
Sp(4), SU(4): F=3/2
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Phase B: the quartetting phase
• Quartetting superfluidity v.s. CDW of quartets (2kf-CDW).
)2/(2 fkd
Kc: the Luttinger parameter in the charge channel.
.2at wins
;2at wins
2
2/32/12/12/3
cLRk
cqt
KN
KO
f
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Phase C: the singlet pairing phase
• Singlet pairing superfluidity v.s CDW of pairs (4kf-CDW).
.at wins
at wins
2
1,4
;2
12/12/12/32/3
cLLRRcdwk
c
KO
K
f
)4/(2 fkd
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• Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
High spin physics with cold atoms
D. M. Stamper-Kurn et al., PRL 80, 2027 (1998); T. L. Ho, PRL 81, 742 (1998); F. Zhou, PRL 87, 80401 (2001); E. Demler and F. Zhou, PRL 88, 163001 (2002);T. L. Ho and S. Yip, PRL 82, 247 (1999); S. Yip and T. L. Ho, PRA 59, 4653(1999).
• Most atoms have high hyperfine spin multiplets.
• High spin fermions: zero sounds and Cooper pairing structures.
F= I (nuclear spin) + S (electron spin).
• Different from high spin transition metal compounds.
eee
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Quintet channel (S=2) operators as SO(5) vectors
)(: 1xy rd
)(: 522 rdyx
i
)(: 2 rd xz i
)(: 3 rd yz
irdrz
)( : 43 22
d
4S
5-polar vector
• Kinetic energy has an obvious SU(4) symmetry; interactions break it down to SO(5) (Sp(4)); SU(4) is restored at U0=U2.
trajectory under SU(2).
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The SU’(4) model: dimensional crossover
• SU’(4) model: 1D dimer order; 2D Neel order.
Jy
Jx
• Competition between the dimer and Neel order.
• No frustration; transition accessible by QMC.
Jy /Jx
1
0
rc
T
dimerrenormalized
classical
quantum disorder
Neel orderC. Wu, O. Motrunich, L. Balents, in preparation .
• SU’(4) model in a rectangular lattice; phase diagram as Jy/Jx.